Simple transitive 2-representations for some 2-subcategories of Soergel bimodules

TL;DR

This paper classifies simple transitive 2-representations of certain 2-subcategories of Soergel bimodules over the coinvariant algebra for Coxeter types B2 and I2(5), revealing new structures and explicit constructions.

Contribution

It provides a complete classification of simple transitive 2-representations for these specific 2-categories, including explicit examples beyond cell 2-representations.

Findings

In type I2(5), all simple transitive 2-representations are cell 2-representations.

In type B2, there is a unique non-cell simple transitive 2-representation.

An explicit construction of the new simple transitive 2-representation in B2 is provided.

Abstract

We classify simple transitive $2$-representations of certain $2$-sub\-ca\-te\-go\-ri\-es of the $2$-category of Soergel bimodules over the coinvariant algebra in Coxeter types $B_2$ and $I_2(5)$. In the $I_2(5)$ case it turns out that simple transitive $2$-representations are exhausted by cell $2$-representations. In the $B_2$ case we show that, apart from cell $2$-representations, there is a unique, up to equivalence, additional simple transitive $2$-representation and we give an explicit construction of this $2$-representation.